-# -# Distributed under the terms of the GNU General Public License (GPL) -# -# This code is distributed in the hope that it will be useful, -# but WITHOUT ANY WARRANTY; without even the implied warranty of -# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU -# General Public License for more details. -# -# The full text of the GPL is available at: -# -# http://www.gnu.org/licenses/ -#***************************************************************************** - -from sage.structure.element import CommutativeAlgebraElement - -def is_CommutativeAlgebraElement(x): - return isinstance(x, CommutativeAlgebraElement)

5. Commutative Algebra - Wikipedia Commutative algebra. In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras.http://facetroughgemstones.com/wikipedia/co/Commutative_algebra.html

# # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #***************************************************************************** from sage.rings.ring import CommutativeAlgebra def is_CommutativeAlgebra(x): return isinstance(x, CommutativeAlgebra)

Contents

Definitions and major results

The notion of commutative ring assumes commutativity of the multiplication operation and usually the existence of a multiplicative identity in addition. The category of commutative rings has

commutative rings as its objects ring homomorphisms as its morphisms; i.e., functions such that is a morphism of abelian groups (with respect to the additive structure of the rings R and R r r r r for all , and R R

Affine Schemes

The theory of affine schemes was initiated with the definition of the prime spectrum of a ring, the set of all prime ideals of a given ring. For curves defined by polynomial equations over a ring

10. Commutative Algebra Commutative Algebra. Home page of A.J. de Jong. The topics we will discuss are Spectrum of a ring, elementary properties, flat and integral ring extensionshttp://www.math.columbia.edu/~dejong/commutative_algebra/

Commutative Algebra

Home page of A.J. de Jong The topics we will discuss are: Spectrum of a ring, elementary properties, flat and integral ring extensions, going up and going down, constructable sets and Chevalley's Theorem, graded modules, Artin-Rees theorem, dimension theory of Noetherian local rings, dimension theory of finitely generated k-algebras and transcendence degree, Hilbert Nullstellensatz. We will attempt to motivate the theory by giving examples from algebraic geometry, but the theorems discussed in the lectures will be theorems of commutative algebra. I will be using the book by Matsumura, Commutative Algebra (Mathematics Lecture Notes Series ; 56), Benjamin-Cummings Pub Co; 2d ed edition (July 1980). It is the book I learned this material from. This book is out of print. On the other hand, you can find all the material covered in any reasonable commutative algebra books. It is a good idea to choose one and stick to it however. Come and find me in my office if you are not sure which books to look at; I have a bunch of them available for you to look at. Another possibility is to use the lecture notes by Professor Robert Friedman from his previous years teaching this course. They are available to registered students through courseworks. It is strongly encouraged to go to the lectures, which are on Tuesday and Thursday 2:40-3:55 in Mathematics 520.

Improve data of this book Suggest different versions or languages of the same book Book details not accurate Add descriptions of this book Add a new cover Report wrong cover for removal EditAllowed tagsbold Italics [0387942696], [ISBN], [ISBN] are actually same versions of the same book Cancel

ARTICLES RELATED TO commutative algebra

commutative algebra: Encyclopedia II - Algebra ring theory - Formal definition Let R be a commutative ring. An R-algebra is a set A which has the structure of both a ring and an R-module in such a way that ring multiplication is an R-bilinear map. Explicity, we must have If A itself is commutative (as a ring) then it is called a commutative R-algebra. Starting with an R-module A, we get an R-algebra by equipping A with an R-bilinear map A  A